Burnt Magsafe connecter

None here can tell you whether the problem is the computer or the ac adapter. We can't see it and we can't evaluate it. You should take both to a qualified shop and have them evaluated. Either way, you have a potential fire hazard on your hands. Until you can be assured that the problem is fixed you should stop using the computer.

I hope you have been backing up the computer. It's about thirteen (!) years old and it seems to be telling you it wants to retire. You should let it. If you don't have a backup, then the best thing to do would be remove the internal drive from the laptop and install it into an external drive case so you can continue to access your files from another computer.

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We want to make sure you reach the right team for help with your device. We recommend that you contact Apple Support directly. Depending on your region, you may be able to reach them here: Get Support.

The Mid 2013 MacBook Airs – there are no Late 2013 ones – do not have hard drives or hard drive bays.

If you buy an external enclosure such as the Other World Computing – OWC Envoy Pro 1A, with the intent of trying to salvage data from your MacBook Air's internal SSD, carefully check your MacBook Air's actual model agains the compatibility information for the enclosure. With old Mac laptops that had "non-serviceable" but removable SSDs, the type of SSD could vary, and you can't assume that the SSDs will work in generic M.2 enclosures.